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#1 2008-05-07 10:37:33

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Fractional calculus

So

But what is

Meet fractional derivatives.

Last edited by JaneFairfax (2008-05-07 10:38:06)


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#2 2008-05-07 10:42:11

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Fractional calculus

Wow! From the article:

(And pi turns up in an unlikely spot again)

Which shows that (in general) if you can think of extending an idea in mathematics, you probably can! What's next ... complex powers?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#3 2008-05-07 19:13:10

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Fractional calculus

¬.¬ this is taking things too far tongue. Although interesting. I would like to see complex powers yes tongue


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#4 2008-07-09 03:38:50

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: Fractional calculus

I've been thinking quite a bit about this (off and on) since it was first posted. When I first read the wiki, it was very much over my head, but now I think I get it. Some further thoughts (please correct me if I'm wrong):

Seeing how:

and it follows that:

for any whole number "n" is it fair to assume:

I think so. But that's easy. For my next trick I'll be using the trig identities:

I think you all probably already see where this is going:

And if "n" is expanded into all real numbers:

Is my logic sound? Does anyone have anymore half derivatives to add?

On a slightly unrelated note, what's the deal with the notation? I've never understood it... why is the second derivative:

and not
?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#5 2008-07-09 03:49:12

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Fractional calculus

When you take the derivative of something, the notation is the change in that something over the change in x.  'd' simply stands for that change.  So it is:

Now we take the derivative of this entire thing.  That is, the change in dy/dx over the change in x.

The last step is because you treat "dx" as a single entity.  At least that's how I always reasoned it.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2008-07-09 07:36:01

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: Fractional calculus

Ok, that makes a bit more sense, I never treated "dx" as a single entity, I always thought it was d(x²) not (dx)²

What about my other work? Any good or am I a little off?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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